Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = (1 − x)³

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

37. y=s√(1-s²) + arccos(s)

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

30. lim (θ → 0) ((1/2)^θ - 1) / θ

Evaluate the integrals in Exercises 53–76.

53. ∫dx/√(9-x²)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

31. y = cos⁻¹(x) - x sech⁻¹(x)

In Exercises 5–8, show that each function is a solution of the given initial value problem.

7. Differential Equation: xy' + y = -sin(x), x>0

Initial condition: y(π/2) = 0

Solution candidate: y = cos(x)/x

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

75. lim (x → ∞) e^(x²) / (x e^x)